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An unknottedness result for noncompact self shrinkers

Alexander Mramor

Abstract

In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.

An unknottedness result for noncompact self shrinkers

Abstract

In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.

Paper Structure

This paper contains 17 sections, 21 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A brief introduction the mean curvature flow
  3. The renormalized mean curvature flow
  4. Proof:
  5. Proof:
  6. The renormalized mean curvature flow with (localized) surgery
  7. Proof:
  8. Proof:
  9. The level set flow of elements of $\Sigma$.
  10. Checking $M_t$ is a forced/renormalized Brakke flow
  11. Proof:
  12. Long term fate of the flow
  13. Proof of Theorem \ref{['thm:unknotted']}
  14. M has an asymptotically conical end
  15. Proof:
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $M^2 \subset \Bbb R{$ R$}^3$ be a two-sided embedded self shrinker which is either compact or has a single, asymptotically conical, end. Then $M$ is topologically standard.

Figures (1)

  • Figure 1: A diagram displaying the possibilities one may encounter at a surgery time for a compact flow in $\Bbb R{$ R$}^3$. High curvature regions are in red and surgery spots are in green. Note in (b) there are two spots along the neck where surgery will be done giving two pairs of caps.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 1.1 of DX
  • Theorem 2.2: Theorem 1.1 of Lu
  • Theorem 2.3: Theorem 7.5 of CY
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 20 more